\begin{section}{Craighero-Gattazzo Surface over a Finite Field of Characteristic 7}

We take $\mathcal{F}$ to be a quintic polynomial describing the Craighero-Gattazzo surface in $\mathbb{P}^3$ over the finite field $\mathbb{Z}_{7}$, which is obtained by taking $\mathcal{F}$ over $\mathbb{Q}$, multiplying through by a common factor to get an expression in integers alone and then reducing $\mathcal{F}$ modulo $7$.

By a similar process, we derive a $7$-adic Taylor expansion for $\mathcal{F}$ near the origin which is given by

\begin{equation}
G = LQ^{2} + 7QF_{3} + 49F_{5},
\end{equation}

where $L = (x + y + z + t)$ is a plane, $Q = (xz + yt)$ is a quadric cone and $F_{3}, F_{5}$ are the first and second order $7$-adic Taylor polynomials of $\mathcal{F}$.

Then we compute the expression for the $7$-adic Taylor expansion, immediately dehomogenize by setting $t = 1$ and eliminate the variable $y$ by setting $y = -xz$. Then we analyze the ``discriminant'' $\nabla = F_{3}^{2} - 4LF_{5}$ of $G$ in this coordinate patch.

Using Macaulay2, we check that

\begin{equation}
\nabla = (-3)(x^3z+x^2z^2+2xz^3-2x^2-xz-z^2)(x^3z^2-3x^2z^3+3x^2z+3xz^2-3x+z).
\end{equation}

\begin{rmk}
This expression for $\nabla$ can be obtained by running the {\it cg-char7} script and then doing $\textrm{factor}(\textrm{sub}(\textrm{disc},\{t=>1,y=>-x*z\})).$
\end{rmk}

\begin{subsection}{Equations for the Tangent to the Discriminant Curve}

Let $f = x^3z+x^2z^2+2xz^3-2x^2-xz-z^2$ and $g = x^3z^2-3x^2z^3+3x^2z+3xz^2-3x+z$. Then we can compute the tangent lines to the discriminant curve by looking at $[f_{x}(x_{0},z_{0})](x-x_{0}) + [f_{z}(x_{0},z_{0})](z-z_{0}) = 0$ (resp., $g, g_{x}, g_{z}$).

\end{subsection} 

\end{section}
